Method of optimizing holographic optical elements

ABSTRACT

A method of designing a holographic optical element (diffractive grating) that transforms a set of waves into another set of waves in order to minimize aberrations in the phase of the output wavefront, includes the steps: (a) define continuous input parameters that characterize the propagation vector components of the incoming wavefront and the desired output wavefront; (b) formulate integral equations for the optimal grating vector components; and (c) solve the integral equations for the optimal grating vector components to minimize the difference between the actual and desired output wavefronts.

FIELD AND BACKGROUND OF THE INVENTION

The present invention relates to a method for optimizing holographicoptical elements, hereinafter referred to as "HOE's".

In an optical system that is designed to operate with monochromatic orquasi-monochromatic illumination sources, it is possible to replace theconventional refractive elements with holographic optical elements(HOE's) that are based on diffractive optics. In general, the HOE'stransform a given set of waves into another set of waves.

The increase in use of monochromatic radiation in complicated opticalsystems that require better optical performance and certain geometricalneeds, has resulted in HOE's becoming very attractive. This isparticularly true for systems operating in the far infrared (IR)radiation, for example 10.6 microns. In such systems, holographicelements that are based on diffractive optics have several advantagesover conventional elements, in that they are thinner, more lightweight,and can perform operations that are impossible by other means.

There are many applications using CO₂ lasers, operating at 10.6 micronswavelength, in which the HOE's are particularly useful. These includelaser material processing, medical surgery, and infrared laser radars.For such applications, since there are no practical recording materialsfor far IR, the HOE's must be formed by using indirect recording. Inpractice, a computer generated mask, representing the grating function,is first plotted with a laser scanner, then reduced in size with opticaldemagnification, and finally recorded as a relief pattern withphotolithographic techniques.

One of the main factors that have hindered the widespread use ofdiffractive elements for far IR radiation is that HOE's have relativelylarge amounts of aberrations. This is because the readout geometries andwavelengths are not identical to the recording geometry and wavelength.In order to minimize the aberrations, it is necessary to useoptimization procedures for designing and recording a holographicelement having a complicated grating function. Several optimizationprocedures have been proposed.

One known optimization procedure is based on numerical iterativeray-tracing techniques. This procedure, however, requires extensivecalculations of ray directions, and the solutions often converge tolocal minima rather than to the desired absolute minimum.

Another known optimization procedure is based on minimizing themean-squared difference of the phases of the actual and desired outputwavefronts. In this procedure, the phase must be defined up to anadditive constant so that the optimization procedure becomes rathercomplicated. It is therefore usually necessary to resort to approximatesolutions.

As a result, such known optimization procedures do not yield an exactsolution except in very specific cases.

OBJECT AND SUMMARY OF THE PRESENT INVENTION

An object of the present invention is to provide a new optimizationprocedure for designing holographic optical elements.

According to the present invention, there is provided a method ofdesigning a holographic diffractive grating that transforms a set ofincident waves into a set of output waves in order to produce outputwaves having minimum aberrations. The method comprises the followingsteps: (a) define continuous input parameters that characterize the setof incident waves and a set of desired output waves with propagationvector components; (b) formulate integral equations for optimal gratingvector components of the holographic diffractive grating to be designed;and (c) solve the integral equations for the optimal grating vectorcomponents to minimize the difference between the propagation vectorcomponents of a set of waves actually emerging from the holographicdiffractive grating and the set of desired output waves.

In the described preferred embodiment, the integral equations are solvedanalytically.

The input parameters may include the direction cosine of the set ofincoming waves, or the location of the input point sources. The incomingwavefront may be from a monochromatic illumination source, such as asource of infrared laser radiation, e.g., a CO₂ laser of 10.6 microns.The incoming wavefront may also be from a quasi-monochromaticillumination source.

In the described preferred embodiment, a computer generated maskrepresenting the grating function is first plotted with a laser scanner,then reduced in size with optical demagnification, and finally recordedas a relief pattern using photolithographic techniques.

As will be shown more particularly below, the optimization procedureaccording to the present invention is based on analytic ray-tracing thatminimizes the mean-squared difference of the propagation vectorcomponents between the actual and the desired output wavefronts. Themean-squared difference of the vector components is defined in such away that the functions involved are continuous.

The design and recording techniques are described below with respect toaspheric low f-number, reflective, off-axis, focussing holographicelements for a readout wavelength of 10.6 microns from a CO₂ laser.These aspheric elements have diffraction-limited spot sizes also forplane waves arising from relatively large incidence angles.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention is herein described, by way of example only, withreference to the accompanying drawings, wherein:

FIGS. 1 and 1a illustrate the readout geometry for a reflective off-axisand on-axis, respectively, holographic focussing element (HFE);

FIGS. 2a, 2b and 2c are spot diagrams for the off-axis HFE, FIG. 2aillustrating the spherical grating function, FIG. 2b illustrating thequadratic grating function, and FIG. 2c illustrating the optimizedgrating function;

FIGS. 3a and 3b illustrate typical etched sections of a modulatedsurface of the HFE, each white line extending over 10 μm in FIG. 3a, andover 100 μm in FIG. 3b;

FIG. 4 illustrates the relative power of the light as a function of thedisplacement of the knife edge at the focussing plane for one inputplane wave at θ_(xi) =30°, θ_(yi) =0° shown by the solid curve, thecorresponding intensity distribution of the focussed spot being shown bythe dashed curved; and

FIG. 5 illustrates the spot size as a function of the distance (d_(out))between the hologram and the measurement plane, for an input plane waveat θ_(xi) =25°, θ_(yi) =0°, the experimental data being depicted by theerror bars, and the solid curve representing the calculated results.

Before describing a preferred embodiment of the invention it will behelpful first to describe the general optimization procedure on whichthe present invention is based.

THE GENERAL OPTIMIZATION PROCEDURE

A HOE is generally described as a diffractive grating that modifies thephase of an incoming wavefront to another output phase. Accordingly, thephase of the output wavefront, φ_(o), for the first diffracted order isgiven by

    φ.sub.o φ.sub.i -φ.sub.h,                      (1)

where φ_(i) is the phase of the input wavefront and φ_(h) is the gratingfunction of the HOE. We found that it is advantageous to exploit thenormalized propagation vectors and grating vector of the holographicelement, rather than the phases.

The normalized propagation vectors, which can be regarded as thedirection cosines of the input (K_(i)) and output (K_(o)) rays, can bewritten as ##EQU1## and the grating vector K_(h), as ##EQU2## where ∇ isthe gradient operator, Λ_(x) and Λ_(y) are the grating spacing in x andy directions, and λ is the readout wavelength. The diffraction relationscan now be written as ##EQU3## Note that K_(x).sbsb.o² +K_(y).sbsb.o²would be less than one so as not to obtain evanescent wavefronts.

The goal when designing HOEs is to transfer input rays intocorresponding output rays that will be optimized for a given range ofinput parameters. The input parameter could, for example, be thedirection cosine of the incoming waves, or the location of the inputpoint sources. For a single specific input parameter it is relativelyeasy to form a HOE that will yield the exact desired output rays.However, for a range of input parameters, it it necessary to optimizethe grating vector so as to minimize the difference between the actualand the desired output rays. The optimization is achieved by minimizingthe mean-squared difference between these two sets of rays.

We shall describe the method in a two dimensional notation. Themean-squared difference of the propagation vector components of theactual and desired output rays includes two scalar equations, and iswritten as

    E.sub.⊥.sup.2 ≡∫∫∫∫[K.sub.⊥.sbsb.d (x,y,a,b)-K.sub.⊥.sbsb.o (x,y,a,b)].sup.2 dadbdxdy,  (7)

where ⊥ denotes the transverse vector components x and y, K.sub.⊥.sbsb.d(x,y,a,b) and K.sub.⊥.sbsb.o (x,y,a,b) are the direction cosines of thedesired and actual output rays, x and y are the space coordinates on theHOE, a and b are the input parameters for x and y coordinatesrespectively. Equation (7) can be expanded, by using Eqs. (4) and (5),to

    E.sub.⊥.sup.2 =∫∫∫∫[K.sub.⊥.sbsb.d (x,y,a,b)-K.sub.⊥.sbsb.i (x,y,a,b)+K.sub.⊥.sbsb.h (x,y)].sup.2 dadbdxdy.                                                 (8)

The optimal grating vector components K_(x).sbsb.h (x,y) andK_(y).sbsb.h (x,y) can be determined by minimizing E_(x) ² and E_(y) ².However, because the integrand is always positive, it is sufficient tominimize a simpler integral, that we denote as e.sub.⊥² (x_(o),y_(o)),

    e.sub.⊥.sup.2 (x.sub.o,y.sub.o)≡∫∫[K.sub.⊥.sbsb.d (x.sub.o,y.sub.o,a,b)-K.sub.⊥.sbsb.i (x.sub.o,y.sub.o,a,b)+K.sub.⊥.sbsb.h (x.sub.o,y.sub.o)].sup.2 dadb,(9)

where x_(o) and y_(o) represent arbitrary coordinates of x and y.Differentiating e.sub.⊥² (x_(o),y_(o)) with respect to K.sub.⊥.sbsb.h(x_(o),y_(o)), setting the result to zero, and noting that the secondderivative of e.sub.⊥² is always positive, yields the optimal gratingvector components, ##EQU4##

We would like to note that, in general, the optimization procedure caninclude the effects of different optimization weighting for each inputparameter, and readout with broad spectral illumination. In such a caseEq. (10) would be generalized to ##EQU5## In this generalized equationW(a) and W(b) are the optimization weighting functions for the inputparameters a and b, where 0≦W(a)≦1 and 0≦W(b)≦1. Also, W.sub.μ (μ)denotes the weighing function for the readout wavelengths; where0≦W.sub.μ (μ)≦1 and μ is the ratio of the readout wavelength over therecording wavelength.

The optimal two dimensional grating function φ_(h) (x,y) can be found,using Eq. (3), by integrating along some arbitrary path to yield##EQU6## where φ_(h) (0,0) can be defined as zero.

For a unique solution, the condition of ∇.sub.⊥ ×K_(h) =0 must befulfilled, where the gradient ∇.sub.⊥ denotes ##EQU7## This conditioncan be written explicitly as ##EQU8## For an on-axis holographicelement, having circular symmetry, as illustrated in FIG. 1a thiscondition is always fulfilled. However, in general, for off-axiselements, this condition is not fulfilled, so that an exact solution forthe grating function φ_(h) (x,y) cannot be found. Nevertheless, it ispossible to obtain approximate solutions. For example, when the off-axisangle is relatively low, it is possible to approximate the gratingfunction by simply adding a linear phase term to the on-axis design.

A more general approximate solution for φ_(h) (x,y) can be derived bynoting that in many cases K_(x).sbsb.h (x,y) has weak dependence on ycoordinate and K_(y).sbsb.h (x,y) has weak dependence on x coordinateover the whole holographic element's area; thus, these two-dimensionalgrating vector components can be approximated by one-dimensionalcomponents [K_(x).sbsb.h (x)]_(app). and [K_(y).sbsb.h (y)]_(app). Here,the approximation is based on minimizing the mean-squared difference ofthe grating vector components between the optimal two dimensionalgrating vector components and the approximate one-dimensional gratingvector components. The mean-squared difference for x component, forexample, is defined as

    e.sub.x '.sup.2 (x.sub.o)≡∫{[K.sub.x.sbsb.h (x.sub.o)].sub.app. -K.sub.x.sbsb.h (x.sub.o,y)}.sup.2 dy,                    (13)

where K_(x).sbsb.h (x,y) is given by Eq. (10). Differentiating e_(x) '²(x_(o)) with respect to [K_(x).sbsb.h (x_(o))]_(app). and setting theresult to zero, yields the approximate grating vector component ##EQU9##The same approximation procedure for the y component yields ##EQU10##

Consequently, the two-dimensional grating function φ_(h) (x,y), can beapproximated by two separated one-dimensional functions using Eq. (3),and Eqs. (14) and (15), to yield ##EQU11## The approximation of Eq. (16)is sufficiently general so as to be valid for higher off-axis angles andlower f number.

DESCRIPTION OF A PREFERRED EMBODIMENT

The operation and parameters of a reflective Holographic FocussingElement (HFE) are described with the aid of the one-dimensionalrepresentation in FIG. 1. Here each input plane wave converges at theoutput plane, to a point whose location corresponds to the angulardirection of the input wave. The aperture of the HFE extends fromcoordinates D₁ to D₂, whereas the width of the input stop aperture is2W, and it extends from coordinates W₂ to W₁. Finally, d_(i) and d_(o)are the distances from the holographic element to the input stopaperture and output plane, respectively.

We shall now describe the procedure for designing the HFE. For thisdesign, it is convenient to let the input parameters a and b, be thedirection cosines of the input plane waves

    a=α=sin θ.sub.x.sbsb.i,                        (17)

    b=β=sin θ.sub.y.sbsb.i,                         (18)

where θ_(x).sbsb.i and θ_(y).sbsb.i are ninety degrees minus the anglesbetween the incident ray and the x and y axes respectively. Furthermore,to simplify the presentation, we shall only deal with the x componentsof the propagation vectors and of the grating vector.

The normalized propagation vector of the input rays is

    K.sub.x.sbsb.i (x,y,a,b)=α.                          (19)

Now, an input plane wave, having a direction cosine α, must betransformed at a distance d_(o) into a spherical wave converging to apoint (α-α_(r))f, where α_(r) =sin(θ_(r))_(x), (θ_(r))_(x) is theoff-axis angle ((θ_(r))_(y) =0), and f is a proportionality constant.Thus, the direction cosines of the desired output rays become ##EQU12##Substituting K_(x).sbsb.i from Eq. (19) and K_(x).sbsb.d from Eq. (20)into Eq. (10), and using the approximation of Eq. (14), yields ##EQU13##

To solve Eq. (21), the limits of integration must be expressed by theupper α₂ (x,y), β₂ (x,y) and lower α₁ (x,y), β₁ (x,y) direction cosinesof the input plane waves that intercept the holographic lens at a point(x,y). It is then possible to solve Eq. (21) directly by numericalmethods, but for an analytic solution some approximations are needed.First, the extreme direction cosines α₁ (x,y) and α₂ (x,y) areapproximated as a function of the x coordinate only, α₁ (x) and α₂ (x);the details of this approximation are given in the Appendix A. Second,the triple integral of Eq. (21) is simplified to only two integrationsby combining the variables β and y into one variable η_(x) through therelation η_(x) =(y-βf)². According to these approximations, Eq. (21)becomes ##EQU14## where η₂.sbsb.x and η₁.sbsb.x are the upper and lowervalues of η_(x). For the HFE geometry shown in FIG. 1, η₁.sbsb.x isequal to zero and η₂.sbsb.x depends on x. Since the dependence on x isnot strong, it is possible to let η₂.sbsb.x be a free constant parameterthat can be exploited for optimizing the optical performance of theholographic element. Such a free constant parameter could partiallycompensate for the earlier approximations.

The solution of Eq. (22) provides the final holographic grating vectorcomponent as ##EQU15## An identical procedure for determining the ycomponent of the grating vector [K_(y).sbsb.h (y)]_(app). * was used,and with Eq. (16), the two-dimensional grating function for the HFE, wasfound.

To evaluate the performance of the optimally designed HFE, we performeda ray-tracing analysis,¹⁹ using Eqs. (4)-(6); the parameters of theelement were chosen as f=60 mm, d_(o) =60 mm W=10 mm, (θ_(r))_(x) =25°,(θ_(r))_(y) =0°, the ranges of angles θ_(x).sbsb.i and θ_(y).sbsb.i were20°<θ_(x).sbsb.i <30° and -5°<θ_(y).sbsb.i <5° so Δθ_(x).sbsb.i=Δθ_(y).sbsb.i =10°. We found that the focussed spots are sufficientlysmall when d_(i) =55 mm, η₂.sbsb.x =23 mm², η₂.sbsb.y =90 mm², andη₁.sbsb.x =η₁.sbsb.y =0 mm². Note that these parameters were so chosenas to enable separation between the reflected zero orders and thediffracted first orders and to prevent the input stop aperture fromblocking the reflected diffraction orders. We also performed aray-tracing analysis for a quadratic grating function, given by##EQU16## as well as for a spherical grating function, given by##EQU17## The geometrical parameters for the quadratic HFE and for thespherical HFE were the same as above, except that for these elementsd_(i) was chosen so as to optimize the focussed spot sizes; i.e., d_(i)=46 mm for the quadratic HFE and d_(i) =39 mm for the spherical HFE.

The results of the ray tracing analysis for the three focussingelements, which do not take into account the diffraction from theaperture, are shown in FIG. 2. They show the spot diagrams for the threelenses as a function of nine discrete input angles of [θ_(x).sbsb.i,θ_(y).sbsb.i ]. As shown in FIG. 2(a), the small central spot diagramfor the spherical lens is essentially ideal because the recording andreadout geometries are identical. However, as the readout input anglesdiffer from the recording angles, the spot diagrams spreadsubstantially. For the quadratic lens, shown in FIG. 2(b), the spread inthe spot diagrams is comparable to that of the spherical lens. Finally,as shown in FIG. 2(c), it is evident that the lens designed according toour optimized procedure is uniformly superior to the other lenses, withrelatively small spot diagrams, over the entire range of input angles.These results demonstrate that the spot sizes for the optimized lens,are uniformly lower than the diffraction limited size, whereas for thespherical and the quadratic lenses the spot sizes are much larger thanthe diffraction limit ##EQU18##

EXPERIMENTAL REALIZATION AND RESULTS

In order to realize the holographic focussing element, the optimizedgrating function was first plotted as a Lee-type binary computergenerated hologram (CGH) having the same parameters as those describedin the preceding section. The amplitude transmittance of the CGH isgiven by

    t.sub.a =U.sub.s [cos(φ.sub.h (x,y))].                 (26)

The term U_(s) is a unit step function defined by ##EQU19## The binaryCGH was plotted with a laser scanner [Scitex Raystar, Response 300],having resolution capabilities of about 10 μm, and recorded directlyonto photographic film. The recorded plot was then demagnified optically(six times) and recorded as a chrome master mask. The information fromthe mask was transferred by contact printing and suitable exposure ontoa glass substrate coated with aluminum and photoresist. After developingthe photoresist, the aluminum was etched and the remaining photoresistwas removed. Finally, in order to obtain a high reflective finalelement, a thin gold layer was vacuum deposited on the etched aluminumlayer.

FIG. 3 shows two electron microscope pictures of a typical etchedsection of the modulated surface of the element, each with a differentmagnification. As a result of the recording and processing, we end upwith a reflective HOE having a relief pattern. In the scalarapproximation, an incident wavefront is multiplied by the reflectanceH(x,y) of the HOE that is described by ##EQU20## where t_(a) is given byEq. (26) and d is the depth of the surface modulation. The relevantfirst diffracted order is then proportional to the desirede^(i)φ.sbsp.h.sup.(x,y) ; by setting ##EQU21## it is possible tomaximize the diffraction efficiency.

The focussing element was tested with a CO₂ laser at a wavelength of10.6 μm. The focussed spot sizes were measured for various input planewaves by using the scanning knife-edge method. Two stepper motors wereused; one for moving the knife-edge and the other for changing thedistance, d_(out), from the holographic element to the measurementplane. FIG. 4 shows a representative result for the relative power andrelative intensity at the focussing plane, as a function of thedisplacement of the knife edge for an input plane wave at θ_(x).sbsb.i=30°, θ_(y).sbsb.i =0°. The relative power is depicted by the solidcurve. Initially the knife edge does not block any of the focussinglight so the total power is high, but as it scans across the focussedspot, it blocks more of the light. The intensity distribution at themeasurement plane was found by taking the derivative of the solid curve,and the result is shown by the dashed curve. The spot sizes weredetermined directly by multiplying the standard deviation of thedistribution by four. The measurements were performed for the entirerange of input angles, 20°<θ_(x).sbsb.i <30°, -5°<θ_(y).sbsb.i <5° andwe found that the spot sizes were uniformly equal to the diffractionlimit, D_(D).L ≃80 μm [f=60 mm, W=10 mm].

FIG. 5 shows the spot size as a function of the distance d_(out) betweenthe hologram and the measurement plane for one input plane wave atθ_(x).sbsb.i =25°, θ_(y).sbsb.i =0°; the experimental data is given bythe error bars. Also shown (the solid curve) are the spot sizescalculated according to the Fresnel diffraction integral. For thesecalculations, we neglected the geometric aberrations of the focussingelement. The diffraction limited spot size of 80 μm is obtained whend_(out) is 60 mm. As evident, the calculated and experimental resultsare in good agreement.

In the realization procedure, there are several factors that candeteriorate the spot sizes of the holographic element; the quantizationof the grating function by the laser scanner, the aberrations of theoptical demagnification system, and the photolithographic process. Notethat for the focussing element described above, the thinnest line of thegrating function, contains only four demagnified pixels of the laserscanner. Nevertheless, these factors did not significantly degrade theperformance of our element, as we realized a diffraction-limitedperformance for the entire range of input angles.

SUMMARY

We have shown that it is possible to design and realize aspheric low fnumber reflective off-axis focussing elements, for far IR radiation,having a diffraction-limited performance over a broad range of incidenceangles. The design method is based on analytic ray tracing and exploitsthe propagation vectors of the waves, so as to allow the realization ofoptimized HOEs. Results of the ray tracing analysis reveal that lensesdesigned according to our design method perform significantly betterthan spherical and quadratic holographic lenses. The necessary asphericgrating functions were realized by using a laser scanner andphotolithographic techniques to form a CGH. These CGH elements weretested in the laboratory, and the experimental results are in goodagreement with our ray-tracing analysis; specifically,diffraction-limited spot sizes were obtained over a relatively largerange of input angles.

APPENDIX A The Expression for the Extreme Direction Cosines α₁ (x) andα₂ (x)

The expressions for the lower [α₁ (x)] and the upper [α₂ (x)] directioncosines, that represent the pupil function, depend on readout geometry,which in our case is shown in FIG. 1.

The lower [α₁ (x)] direction cosine is ##EQU22## when

    α.sub.1 (x)>α.sub.min,

    otherwise

    α.sub.1 (x)=α.sub.min.                         (2A)

In Eq. (1A), Δy is defined by Δy≡y_(stop) -y, where y_(stop) representsthe input stop aperture coordinate, and y the coordinate at the hologramplane. For a one-dimensional grating function Δy=0, whereas for atwo-dimensional grating function (approximated by two separatedone-dimensional grating functions), Δy is chosen to yield minimum α₁(x). The direction cosine α_(min) is for the plane wave having thelowest angular direction while still completely intercepting thehologram.

The upper [α₂ (x)] direction cosine is ##EQU23## when

    α.sub.2 (x)<α.sub.max,

    otherwise

    α.sub.2 (x)=α.sub.max.                         (4A)

For a one-dimensional grating function Δy=0, whereas for atwo-dimensional grating function Δy is chosen to yield maximum α₂ (x).The direction cosine α_(max) is for the plane wave having the highestangular direction while still completely intercepting the hologram.

What is claimed is:
 1. A method of designing a holographic diffractivegrating that transforms a set of incident waves into a set of outputwaves in order to produce output waves having minimum aberrations,comprising the steps:(a) defining continuous input parameters thatcharacterize the set of incident waves and a set of desired output waveswith propagation vector components; (b) formulating integral equationsfor optimal grating vector components of the holographic diffractivegrating to be designed; and (c) solving said integral equations for theoptimal grating vector components to minimize the difference between thepropagation vector components of a set of waves actually emerging fromthe holographic diffractive grating and the set of desired output waves.2. The method according to claim 1, wherein said integral equations aresolved analytically.
 3. The method according to claim 1, wherein saidintegral equations are solved numerically.
 4. The method according toclaim 1, wherein the input parameters include the direction cosines ofthe set of incident waves.
 5. The method according to claim 1, whereinthe input parameters include locations of input point sources.
 6. Themethod according to claim 1, wherein the set of incident waves is from amonochromatic illumination source.
 7. The method according to claim 6,wherein the monochromatic illumination source is a laser.
 8. The methodaccording to claim 7, wherein said laser outputs radiation of 10.6microns.
 9. The method according to claim 1, wherein the set of incidentwaves is from a quasi-monochromatic illumination source.
 10. The methodaccording to claim 1, wherein the holographic diffractive grating is anon-axis holographic element having circular symmetry.
 11. The methodaccording to claim 1, wherein the holographic diffractive grating is anoff-axis holographic diffractive grating, the optimal grating vectorcomponents being obtained by certain approximations including oneapproximation based on minimizing the mean squared difference of thegrating vector components between the optimal two-dimensional gratingvector components and the approximate one-dimensional grating vectorcomponents.
 12. The method according to claim 1, wherein a mask isgenerated by a computer according to the above steps and is firstplotted with a laser scanner, then reduced in size with opticaldemagnification, and finally recorded as a relief pattern by using aphotolithographic technique.